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Properties

Label 2-309-309.137-c0-0-0
Degree $2$
Conductor $309$
Sign $0.289 - 0.957i$
Analytic cond. $0.154211$
Root an. cond. $0.392697$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.273 + 0.961i)3-s + (0.445 + 0.895i)4-s + (−0.0505 − 0.544i)7-s + (−0.850 − 0.526i)9-s + (−0.982 + 0.183i)12-s + (0.0822 + 0.887i)13-s + (−0.602 + 0.798i)16-s + (−0.510 − 1.79i)19-s + (0.538 + 0.100i)21-s + (0.739 + 0.673i)25-s + (0.739 − 0.673i)27-s + (0.465 − 0.288i)28-s + (1.18 − 1.56i)31-s + (0.0922 − 0.995i)36-s + (−0.181 + 0.0339i)37-s + ⋯
L(s)  = 1  + (−0.273 + 0.961i)3-s + (0.445 + 0.895i)4-s + (−0.0505 − 0.544i)7-s + (−0.850 − 0.526i)9-s + (−0.982 + 0.183i)12-s + (0.0822 + 0.887i)13-s + (−0.602 + 0.798i)16-s + (−0.510 − 1.79i)19-s + (0.538 + 0.100i)21-s + (0.739 + 0.673i)25-s + (0.739 − 0.673i)27-s + (0.465 − 0.288i)28-s + (1.18 − 1.56i)31-s + (0.0922 − 0.995i)36-s + (−0.181 + 0.0339i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7631928802\)
\(L(\frac12)\) \(\approx\) \(0.7631928802\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.273 - 0.961i)T \)
103 \( 1 + (0.982 - 0.183i)T \)
good2 \( 1 + (-0.445 - 0.895i)T^{2} \)
5 \( 1 + (-0.739 - 0.673i)T^{2} \)
7 \( 1 + (0.0505 + 0.544i)T + (-0.982 + 0.183i)T^{2} \)
11 \( 1 + (-0.445 - 0.895i)T^{2} \)
13 \( 1 + (-0.0822 - 0.887i)T + (-0.982 + 0.183i)T^{2} \)
17 \( 1 + (-0.0922 - 0.995i)T^{2} \)
19 \( 1 + (0.510 + 1.79i)T + (-0.850 + 0.526i)T^{2} \)
23 \( 1 + (-0.445 + 0.895i)T^{2} \)
29 \( 1 + (-0.739 - 0.673i)T^{2} \)
31 \( 1 + (-1.18 + 1.56i)T + (-0.273 - 0.961i)T^{2} \)
37 \( 1 + (0.181 - 0.0339i)T + (0.932 - 0.361i)T^{2} \)
41 \( 1 + (-0.739 + 0.673i)T^{2} \)
43 \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.850 - 0.526i)T^{2} \)
59 \( 1 + (0.982 + 0.183i)T^{2} \)
61 \( 1 + (0.890 + 0.811i)T + (0.0922 + 0.995i)T^{2} \)
67 \( 1 + (0.156 - 1.69i)T + (-0.982 - 0.183i)T^{2} \)
71 \( 1 + (-0.739 + 0.673i)T^{2} \)
73 \( 1 + (1.12 - 0.435i)T + (0.739 - 0.673i)T^{2} \)
79 \( 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2} \)
83 \( 1 + (0.982 - 0.183i)T^{2} \)
89 \( 1 + (0.602 + 0.798i)T^{2} \)
97 \( 1 + (-1.09 + 0.995i)T + (0.0922 - 0.995i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.71980954207606658947553872599, −11.31924994672049810443738468900, −10.37934131851159296801505144448, −9.253153411220694120019938286183, −8.501257084631932995202929547701, −7.18564125034639440843469713546, −6.38077104883921208794446619495, −4.78573209251387497396351797959, −3.95102814534616506698251920290, −2.71021094173457112650042824003, 1.52158454278099003830526690324, 2.88889484988840909034830212359, 5.04832255034927325455607577889, 5.97389935245869734328708555491, 6.62826029749905741015203715649, 7.86748721778766670783067706057, 8.725752920927335798875487881323, 10.19854275540043220622845735551, 10.70513575954049904860216082026, 11.98522304907917298917362107916

Graph of the $Z$-function along the critical line